Reinforce crawl space floor structure - beam under exterior wall

[Husåhem]

We are going to install a stove and need to reinforce the floor joists. The stove will be placed by an exterior wall, on the ground floor at one end of the gable side of the house (1.5-story villa built in 1981).

The crawl space foundation is constructed with three large concrete foundations running along the long side of the house; one under each of the long exterior walls and one in the middle of the house. On the gable sides, there are also concrete foundations, but these are somewhat smaller than those on the long sides.

The initial plan was to support the floor joist that is about 60cm out from the gable exterior wall (joist no. 2 in sequence). Add blocking between the floor joist under the exterior wall and joist no. 2. Then pour a pad with an adjustable post bracket and place a vertical joist as reinforcement under joist no. 2.

When I went into the crawl space and opened the asfaboard, I saw that the joist under the gable exterior wall is not in contact with the crawl space wall. On the long sides of the house, the exterior wall sill rests on the concrete foundation, but not on the gable side. On the gable side, the joist seems to only rest on the larger long side beams and the middle beam.

Question:
Is it necessary to also reinforce the joist that lies under the gable side exterior wall?

Is it possible to insert Syllpapp in the small gap (about 8-9 mm) between the top of the concrete wall and the asfaboard that lies under the current joist? (This way, the joist cannot sink) That is, can the concrete wall take the load, or is its purpose only to hold the crawl space together?

 

[Husåhem]

Adding an additional question:
Should one only screw in noggins between floor joists in a "cold" floor structure facing a crawl space, or can brackets be used? (angle brackets, etc.)
I'm thinking about the thermal bridges you risk getting with brackets.